# Schrijver’s SDP Bound for Network Codes

This is a small report on a failed project — obtaining semidefinite programming bounds on constant dimension network codes. But let us start with some context …

A network code consists of a set of subspaces in $\mathbb{F}_q^n$. It is a code, so we want to maximize the distance between subspaces (or increase the code’s cardinality or rate). The most reasonable metric for this is the subspace metric in which two subspaces $U$ and $W$ have distance $\dim(U+W) - \dim(U \cap W)$, that is the shortest distance between $U$ and $W$ in the subspace lattice. These codes are used in random linear network coding, that is a method for the many-to-many transmission of data in a network which is faster than multiple one-to-one connections. As in classical coding theory, one area of research on those subspace codes is concerned with bounding the maximal size of a code with minimum distance $d$ in $\mathbb{F}_q^n$. Notice that one can interpret $q=1$ as a field with one element as is explained here by Peter Cameron. Then it corresponds to classical coding theory. (more…)